ACT Preparation / Unit 8 ACT 0710 worked example, 60 to 90 seconds
Factoring a Trinomial With a Leading Coefficient
The idea
To factor ax² + bx + c when a is not 1, work in five steps. First, take out any common factor. Second, multiply a by c. Third, list factor pairs of ac until one pair sums to b. Fourth, rewrite the middle term as two terms using that pair. Fifth, factor each half by its own common factor, then factor out the shared binomial. Check by multiplying the factors back.
In practice
Worked example: factor 6x² − 7x − 3. Step two: ac = 6 × −3 = −18. Step three: −9 and 2 multiply to −18 and add to −7. Step four: 6x² − 9x + 2x − 3. Step five: 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3). Check: (3x + 1)(2x − 3) = 6x² − 9x + 2x − 3 = 6x² − 7x − 3.
Where people slip
When the second pair's common factor is negative, take the negative sign out with it, so both pairs show the same binomial: 4x² − 4x − 3x + 3 = 4x(x − 1) − 3(x − 1).
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